A note on invariance in three-mode factor analysis

Previous results of the application of Lawley's selection theorem to the common factor analysis model are extended to a revision of Tucker's three-mode principal components model. If the regression of the three-mode manifest variates on variates used to select subpopulations is both linear and homoscedastic, the two factor pattern matrices, the core matrix, and the residual variance-covariance matrix in the three-mode model can all be assumed to be invariant across subpopulations. The implication of this finding for simple structure is discussed.     

Robustness of the maximum-likelihood estimation procedure in factor analysis

In order to determine the robustness of the maximum-likelihood estimation procedure, random variates from six distributions were generated independently on a high speed computer and then used to represent the common and specific factors in a factor analysis model in which the coefficients of these factors had been specified. Using Lawley's approximatex ² statistic in evaluating the estimates obtained, the estimation procedure is found to be insensitive to changes in the distributions considered.This study was conducted while the authors were members of the Statistical Laboratory at Iowa State University. All computations were done at Iowa State on an IBM 7074.     

Determination of the Regression Coefficients and Their Associated Standard Errors in Hierarchical Regression Analysis

Hierarchical regression analysis is potentially a very useful statistical technique for establishing the significance of sets of predictor variables. However, when a hierarchical analysis which is based on theory is performed, some estimation procedures for the regression coefficients and their associated standard errors are potentially inappropriate. Specifically, the hierarchical regression equations, the incremental or hierarchical tests, and the parameter estimation of this procedure may not correspond. This problem is investigated by the development of four approaches (simultaneous, stagewise, orthogonal, and hierarchical) of estimation to the analysis. For each method, regression, coefficient estimators and their standard errors are determined. By comparison of these approaches, the use of orthogonalized sets of predictor variates or a modification to a series of simultaneous analyses are recommended as the most sensible technique for a theory driven hierarchical analysis.     

随机截距因子分析模型在控制条目表述效应中的应用

本研究用中文修订版罗森博格自尊量表(RSES-R)考察随机截距因子分析模型在控制条目表述效应时的表现。用RSES-R和过分宣称问卷组成的量表调查621名中学生。结果表明,随机截距模型在建模时,拟合指数良好、因子方差与负荷合理,自尊因子分与RSES-R总分有极高相关,表明该模型能有效分离RSES-R得分的特质与表述效应。分离的表述效应因子分与受测者的自我提升水平具有显著但较弱的相关,表明表述效应与自受测者的社会赞许性有共同的成分。     

Joint Maximum Likelihood Estimation for High-Dimensional Exploratory Item Factor Analysis

Joint maximum likelihood (JML) estimation is one of the earliest approaches to fitting item response theory (IRT) models. This procedure treats both the item and person parameters as unknown but fixed model parameters and estimates them simultaneously by solving an optimization problem. However, the JML estimator is known to be asymptotically inconsistent for many IRT models, when the sample size goes to infinity and the number of items keeps fixed. Consequently, in the psychometrics literature, this estimator is less preferred to the marginal maximum likelihood (MML) estimator. In this paper, we re-investigate the JML estimator for high-dimensional exploratory item factor analysis, from both statistical and computational perspectives. In particular, we establish a notion of statistical consistency for a constrained JML estimator, under an asymptotic setting that both the numbers of items and people grow to infinity and that many responses may be missing. A parallel computing algorithm is proposed for this estimator that can scale to very large datasets. Via simulation studies, we show that when the dimensionality is high, the proposed estimator yields similar or even better results than those from the MML estimator, but can be obtained computationally much more efficiently. An illustrative real data example is provided based on the revised version of Eysenck’s Personality Questionnaire (EPQ-R).

     

Robustness of normal theory methods in the analysis of linear latent variate models

The structure of the covariance matrix of sample covariances under the class of linear latent variate models is derived using properties of cumulants. This is employed to provide a general framework for robustness of statistical inference in the analysis of covariance structures arising from linear latent variate models. Conditions for normal theory estimators and test statistics to retain each of their usual asymptotic properties under non-normality of latent variates are given. Factor analysis, LISREL and other models are discussed as examples.     

THE EFFECT OF FACTOR SCORES,GUTTMAN SCORES,AND SIMPLE SUM SCORES ON THE SIZE OF F RATIOS IN AN ANALYSIS O F VARIANCE DESIGN

Many questions in the social sciences reduce to a comparison of mean values across groups in a classical analysis of variance F test. Often the original data my come from a set of items in a questionnaire or personality inventory. When this occurs, some sort of data reduction, combining of items, or scaling procedure is first performed before the hypothesis of no difference in mean values across groups can be made. In many cases, this problem causes undue concern t0 a researcher because the effect of the scoring procedure on the distribution of F is not clear. To help solve this problem, this study was undertaken to investigate whether the method used to calculate scores has any effect on the magnitude of the F ratio in an analysis of variance, for, if it were shown that no statistical difference existd, then a researcher would have some justification for showing the procedure having minimal messes. On the other hand, if statistical differences were b arise because of the kind d scaling procedure employed, then a researcher would have to be more cautious in his choice. For this empirical investigation, Guttman, Saaotor, and simple sum scores were generated using item responses from a large pool of high school seniors. No difference in scoring method was detected when the F ratios resulting from each of the three scoring methods were analyzed. This suggests that, for chin analyses, a simple sum score may be as effective as mres derived by more complicated methods.     

Redundancy analysis an alternative for canonical correlation analysis

A component method is presented maximizing Stewart and Love's redundancy index. Relationships with multiple correlation and principal component analysis are pointed out and a rotational procedure for obtaining bi-orthogonal variates is given. An elaborate example comparing canonical correlation analysis and redundancy analysis on artificial data is presented.A Fortran IV program for the method of redundancy analysis described in this paper can be obtained from the author upon request.     

探索性结构方程建模(ESEM):EFA和CFA的整合

探索性结构方程建模(ESEM)是在测量模型部分使用了类似于EFA模型的SEM.作为一种高级统计方法,ESEM整合了EFA和CFA两种因子分析方法的功能和优点.通过ESEM,研究者既可以灵活地探索因子结构,又可以系统地验证因子模型,为潜变量的关系分析提供更适宜的测量模型.ESEM已经在某些社科领域的研究中得到应用,是一种值得推介的因子分析方法.ESEM的具体应用问题,例如因子旋转方法的选用、测验信度评价等,仍有待探讨.     

Instrumental variables in factor analysis

The factor analysis model is rewritten as a system of linear structural relations with errors in variables. The method of instrumental variables is applied to this revised form of the model to obtain estimates of the factor loading matrix. The relation between this method and interbattery analysis, proportional profile analysis, and canonical factor analysis is pointed out. In addition, an estimation procedure based on replicated sampling different from proportional profile analysis is given.     

Storage-retrieval analysis of acoustic similarity

A statistical procedure that separates storage from retrieval was used to study the acoustic similarity effect as a function of retention interval in the Brown-Peterson paradigm. Both storage and retrieval components showed reliable and independent changes with retention interval, but only the storage component was affected by acoustic similarity. Hence, acoustic similarity affects trace durability and retrieval plays no essential role in the similarity effect. This finding is inconsistent with the address hypothesis (cf. Baddeley, 1968). It is argued that acoustic similarity induces subjects to encode the target item in a confused fashion, particularly in regard to order information.     

Order effects in belief updating with consistent and inconsistent evidence

The current study tests for the presence of differential order effects in evaluation tasks with consistent and inconsistent evidence as predicted by the Hogarth and Einhorn (1992) belief-adjustment model. The results, based on both between-subjects and within-subjects experiments, demonstrate that there were significant recency effects with inconsistent evidence as predicted, larger recency effects when the inconsistent evidence was farther apart in subjective value as predicted, and significant recency effects even when subjects were given training designed to both help them understand the task as completely as possible and to be better able to assess the pieces of evidence. By including a within-subjects design, we were able to demonstrate that the difference in subjective value between two pieces of evidence is the primary factor influencing the magnitude of the recency effect, regardless of whether the evidence is consistent or inconsistent. This latter finding is unique and contrary to previous research and theory.     

Visual search for emotional expressions: Effect of stimulus set on anger and happiness superiority

Prior reports of preferential detection of emotional expressions in visual search have yielded inconsistent results, even for face stimuli that avoid obvious expression-related perceptual confounds. The current study investigated inconsistent reports of anger and happiness superiority effects using face stimuli drawn from the same database. Experiment 1 excluded procedural differences as a potential factor, replicating a happiness superiority effect in a procedure that previously yielded an anger superiority effect. Experiments 2a and 2b confirmed that image colour or poser gender did not account for prior inconsistent findings. Experiments 3a and 3b identified stimulus set as the critical variable, revealing happiness or anger superiority effects for two partially overlapping sets of face stimuli. The current results highlight the critical role of stimulus selection for the observation of happiness or anger superiority effects in visual search even for face stimuli that avoid obvious expression related perceptual confounds and are drawn from a single database.     

The structure of the Hospital Anxiety and Depression Scale: Theoretical and methodological considerations

The Hospital Anxiety and Depression Scale (HADS; Zigmond - Snaith, 1983) is widely used; however, its factor structure is unclear, with studies reporting differing unidimensional, two-factor and three-factor models. We aimed to address some key theoretical and methodological issues contributing to inconsistencies in HADS structures across samples. We reviewed existing HADS models and compared their fit using confirmatory factor analysis (CFA). We also investigated methodological effects by comparing factor structures derived from Rasch and Principal Components Analysis (PCA) methods, as well as effects of a negative wording factor. An Australian community-dwelling sample consisting of 189 females and 158 males aged 17–86 (M = 35.73, SD = 17.41) completed the 14-item HADS. The Rasch Analysis, PCA and CFA all supported the original two-factor structure. Although some three-factor models had good fit, they had unacceptable reliability. In the CFA, a hierarchical bifactor model with a general distress factor and uncorrelated depression and anxiety subscales produced the best fit, but the general factor was not unidimensional. The addition of a negative wording factor improved model fit. These findings highlight the effects of differing methodologies in producing inconsistent HADS factor structures across studies. Further replication of model fit across samples and refinement of the HADS items is warranted.     

反应风格的测量与统计控制

反应风格是共同方法偏差的主要来源之一。本文首先讨论反应风格的定义和类型,梳理其危害,认为反应风格能使测验分数出现偏差,影响测验信效度分析和变量关系分析,有必要控制其危害。然后介绍了常用的反应风格测量方法,包括计数法和模型法两大类,对测量方法的选择给出了建议,在此基础上,就如何结合反应风格的测量方法与残差回归法、偏相关法来控制反应风格危害给出建议。     

反应风格的测量与统计控制

反应风格是共同方法偏差的主要来源之一。本文首先讨论反应风格的定义和类型,梳理其危害,认为反应风格能使测验分数出现偏差,影响测验信效度分析和变量关系分析,有必要控制其危害。然后介绍了常用的反应风格测量方法,包括计数法和模型法两大类,对测量方法的选择给出了建议,在此基础上,就如何结合反应风格的测量方法与残差回归法、偏相关法来控制反应风格危害给出建议。     

Analysis of Variance of Multiply Imputed Data

As a procedure for handling missing data, Multiple imputation consists of estimating the missing data multiple times to create several complete versions of an incomplete data set. All these data sets are analyzed by the same statistical procedure, and the results are pooled for interpretation. So far, no explicit rules for pooling F tests of (repeated-measures) analysis of variance have been defined. In this article we outline the appropriate procedure for the results of analysis of variance (ANOVA) for multiply imputed data sets. It involves both reformulation of the ANOVA model as a regression model using effect coding of the predictors and applying already existing combination rules for regression models. The proposed procedure is illustrated using 3 example data sets. The pooled results of these 3 examples provide plausible F and p values.     

On equivariance and invariance of standard errors in three exploratory factor models

Current practice in factor analysis typically involves analysis of correlation rather than covariance matrices. We study whether the standardz-statistic that evaluates whether a factor loading is statistically necessary is correctly applied in such situations and more generally when the variables being analyzed are arbitrarily rescaled. Effects of rescaling on estimated standard errors of factor loading estimates, and the consequent effect onz-statistics, are studied in three variants of the classical exploratory factor model under canonical, raw varimax, and normal varimax solutions. For models with analytical solutions we find that some of the standard errors as well as their estimates are scale equivariant, while others are invariant. For a model in which an analytical solution does not exist, we use an example to illustrate that neither the factor loading estimates nor the standard error estimates possess scale equivariance or invariance, implying that different conclusions could be obtained with different scalings. Together with the prior findings on parameter estimates, these results provide new guidance for a key statistical aspect of factor analysis.We gratefully acknowledge the help of the Associate Editor and three referees whose constructive comments lead to an improved version of the paper. This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017 and by the University of North Texas Faculty Research Grant Program.